Problem 24 - Elementary Algebra

Question

For the following problems, observe the equations and write the relationship being expressed. $$ x=5 y^{3}+2 y+6 $$

Step-by-Step Solution

Answer

Question: Explain in words the relationship between variables x and y in the given equation: $$x = 5y^3 + 2y + 6$$. Answer: In the given equation, x is a function of y, which means x depends upon the value of y. Specifically, x is equal to the sum of a cubic term of y ($$5y^3$$), a linear term of y ($$2y$$), and a constant term ($$6$$). As the value of y changes, the value of x will also change accordingly based on the equation $$x = 5y^3 + 2y + 6$$.

Step 1: Identify the variables involved in the equation

In this equation, there are two variables: x and y. This equation shows a relationship between x and y.

Step 2: Understand the terms associated with the variable y

In the given equation, there are three terms associated with y: $$5y^3$$, $$2y$$, and $$6$$ (which is a constant term). Each term has a different degree of y. The first term, $$5y^3$$, is a cubic term (degree 3) as y is raised to the power of 3. The second term, $$2y$$, is a linear term (degree 1) as y is raised to the power of 1.

Step 3: Describe the equation using the variable relationships

Using the terms identified in Step 2, we can describe the relationship between x and y as follows: x is equal to the sum of a cubic term of y ($$5y^3$$), a linear term of y ($$2y$$), and a constant term ($$6$$). In simpler terms, x is a function of y, given by the equation $$x = 5y^3 + 2y + 6$$.

Key Concepts

Cubic Polynomial

A cubic polynomial is a type of polynomial where the highest power of the variable is three. In simple terms, it's an equation that might look something like:

• x = ay^3 + by^2 + cy + d

Here, the term with the variable raised to the power of three is what we call the 'cubic term,' and it's the defining feature of a cubic polynomial. In the original exercise we looked at, the equation is:

• x = 5y^3 + 2y + 6

This equation is a cubic polynomial because it includes the cubic term \(5y^3\). The other terms in the equation, \(2y\) and \(6\), are lesser degree terms—the linear and constant terms, respectively. What makes cubic polynomials interesting is how their graphs behave. They can have one or more turning points, showing changes in direction, unlike simpler polynomials like linear or quadratic ones.

Variable Relationships

Variable relationships describe how changes in one variable affect changes in another. In our example, the equation \(x = 5y^3 + 2y + 6\) connects the variables \(x\) and \(y\). This shows that \(x\) depends on \(y\). Let's break it down:

• The cubic term \(5y^3\) suggests a significant change in \(x\) for any small change in \(y\). The effect grows rapidly as \(y\) increases or decreases.
• The linear term \(2y\) implies that \(x\) changes at a constant rate as \(y\) changes.
• The constant term \(6\) is independent of \(y\) and doesn't change when \(y\) does.

By understanding these relationships, you can predict how the value of \(x\) will change when \(y\) is altered—a foundational concept in calculus and various scientific models.

Function of a Variable

A function of a variable explains how one quantity completely depends on another. In the given equation, \(x = 5y^3 + 2y + 6\), \(x\) is expressed as a function of \(y\). This means:

• For every possible value of \(y\), there is a corresponding unique value of \(x\).
• The rules, expressed in terms of \(y\), define \(x\). Think of it like a recipe, where \(y\) is an ingredient list, and \(x\) is the resulting dish.

In terms of the exercise, understanding this function helps clarify how the entire equation builds to describe \(x\). Functions of a variable are critical in mathematics as they allow us to model real-world situations, helping find solutions to complex problems through understanding how variables are interconnected.